Spinors pure-spin

In this section I will outline the different methods that have been used and are currently used for the computation of parity violating effects in molecular systems. First one-component methods will be presented, then four-component schemes and finally two-component approaches. The term one-component shall imply herein that the orbitals employed for the zeroth-order description of the electronic wavefunction are either pure spin-up spin-orbitals or pure spin-down spin-orbitals and that the zeroth-order Hamiltonian does not cause couplings between the two different sets ( spin-free Hamiltonian). The two-component approaches use Pauli bispinors as basic objects for the description of the electronic wavefunction, while the four-component schemes employ Dirac four-spinors which contain an upper (or large) component and a lower (or small) component with each component being a Pauli bispinor. 

These combinations therefore give pure bonding and antibonding diatomic molecular spinors, which also have pure spin. For homonuclear diatomic molecules, the spinors have a definite inversion symmetry a = ofg, a = a , ti = 7i , n = Ttg, and so on. The Kramers partners of these spinors are found by interchanging the spin parts. The first one is... 

Since the spinors formed from the two spin-orbit components have the same >, we may take a linear combination of them. One way of combining them results in pure-spin spinors ... 

These combinations are now either totally bonding or totally antibonding, thanks to the fact that one atomic spinor has two components of different sign while the other has the same sign for the two components. The molecular spinors are not symmetric the a part is skewed towards one atom and the ir towards the other. Linear combinations of these spinors will return pure-spin spinors as before. The numerical factors for the a and n orbitals have not been included in these spinors because they have no precise meaning here. 

All electrons, protons and neutrons, the elementary constituents of atoms, are fermions and therefore intrinsically endowed with an amount h/2 of angular momentum, known as spin. Like mass and charge, the other properties of fermions, the nature of spin is poorly understood. In quantum theory spin is treated purely mathematically in terms of operators and spinors, without physical connotation. 

To summarise, the pure valence correlation (double excitations) tends to lower the spin-orbit splitting while the valence spinors relaxation (single excitations) tends to increase the splitting. Concerning the role of the core orbitals, the whole core polarisation, core-core and core-valence correlations taken into account via a semi-empirical CPP tend to enhance this splitting. ... 

Within the Hartree method, the electronic spin does not appear explicitly except for the fact that no more than two electrons may go into a single orbital. The existence of the Pauli exclusion principle, however, needs to be accounted for in order to go beyond the Hartree method, and that is what the Hartree-Fock method [120] is all about. We first formulate an arbitrary three-dimensional orbital for electron i by writing it as the product of a purely space-dependent part and a spin function (spinor), a or characterizing spin-up or spin-down electron, for example (pi Xi) = i(ri)iXi here we use x to indicate a variable which includes both space (r) and spin (a). A Hartree-like product wave function between two one-electron wave functions and 2 could then be written as... 

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